Fall 2016 Spring 2017

Transcription

1 BE 603: Partial differential equations (Spring 2017) Instructor: TA: Andy Fan TBD Class: Time is TBD (Spring 2016 only) Recitation: Should be on Fri afternoons (Spring 2016 only) Office Hours: By appointment, preferably after lunchtime! Office: 44 Cummington St, Room 707 Course documents on Blackboard: Focused on parabolic and elliptical PDEs: Since transport phenomenon is prevalent in the field of biomedical engineering (and we only have 14 full lectures), we will focus our attention on analyzing various forms of Laplace s and Poisson s equations (elliptical), and also, the parabolic PDE regime of the diffusion / heat equation where diffusion dominates over convection. Enrollment: All graduate students or senior undergraduates (need department approval) are welcome! Prerequisites for PDEs (BE 603) Previous exposure to at least 1 of these 2 orthogonal functions from BE 602 (ODEs module): o Legendre polynomials o Bessel functions Have a good handle on undergraduate vector calculus (Divergence theorem, Stoke s theorem) Have seen Fourier series Have seen finite-difference modeling of PDEs for anisotropic medium (from the linear algebra module) Prior exposure to Laplace and Fourier Transforms will help! If you need a refresher on Sturm-Liouville problems, Legendre polynomials, Bessel functions, and Laplace / Fourier transforms, or finite-difference approximations to differential equations, I will have pre-taped video lectures that you can watch at your leisure. This allow us to concentrate on solving PDE-centric problems during lectures, thus avoiding the class being bogged down on rehashing math topics covered in either BE 601, 602, and undergraduate signals & systems courses. Fall 2016 Spring 2017 BE 601: Linear algebra BE 604: Statistical & numerical methods BE 602: Ordinary differential equations BE 603: Partial differential equations

2 Homework and Friday recitations: Problem sets will be handed out usually on Wednesdays and will be due on the following Friday during recitation. During Friday recitations, we will discuss any questions regarding both the previous and next problem sets. Matlab: Since matrix-dependent elements will be in no short supply in this class, we will adopt Matlab as the standard software from which all course materials, homeworks, and take-home tests will be analyzed with. Textbooks: There are no required textbooks for BE 603. Please see pages 4 5 for more explanations! Exams: Take-home exam at the end of each module Duration = 1 week Grading: The breakdown per module is: 80% homeworks (~6 per module) + participation during recitation 20% from take-home exam (1 per module)

3 PDE syllabus: (1 st half of Spring 2016) Lectures (theme) 1-3 Topics Key concepts Applications Separation of variables Segregation of nonhomogeneous BC using steady-state (Type I) vs. transient (Type II) rearrangements The basics: 1D Separation of variables Direct link between ODEs, the eigenvalue problem Ax = x, and Sturm-Liouville eqs Review previously-seen concepts in generalized Fourier series 1D transient and steadystate solutions for heat transfer in solids Pure diffusion problems without convection 4 + rec. +Convection + Reactant loss Nernst-Planck equation 1D transient solutions (with loss terms) Using exponential transformations to remove convection loss terms Neurons (cable theory) Conductive + convective heat transfer and mass transport (no radiation) 5 2D Laplace + Heat 2D Laplace s Eq (Steady-state) 2D Heat Equation (Transient) 2D transient solutions (Rectangular) Generalized Fourier series concepts (sines / cosines; sinh/cosh; exp, etc.) Using linear superposition to construct solutions for PDEs with complicated BCs 2D heat transfer + electrostatics Irrotational flow of inviscid fluids rec 1D + 2D heat (parabolic) 2D Laplace / Poisson (elliptical) 1D and 2D anisotropic FTCS (explicit) 1D anisotropic Crank- Nicolson (implicit) Peaceman-Rachford (ADI schemes for 2D parabolic) Numerical solutions interlude Extending the stead-state conductance matrix formulations from the linear algebra module to time-varying parabaolic PDE problems (diffusion > convection) Choices in approximating convection terms: Center-difference vs. upwind 2D steady-state and transient models for Ion / mass transport and heat transfer Random walk and Brownian motion 3D steady-state (Spherical / Cylindrical) Transport in cylindrical tubes D Laplace s 3D Laplace s equation (steady-state) Generalized Fourier series concepts (Associated Legendre / Bessel functions) Interior / exterior / annular Dirichlet problems in degenerate, 2D polar coords Bounded vs. unbounded 3D diffusion Extending 2D FCTS to polar coordinates (Cauchy) Fourier / Laplace transform solutions Green s functions Method of images Integral transforms solutions / Green s functions Using symmetry, superposition, the 2 types of integral transforms, and the Green s functions to construct PDE solutions Neurons (cable theory) Diffusion-convection equations with semi-infinite BCs Economics (Black-Scholes)

4 BE 603: Partial differential equations (reading list) Recommended textbook: For introduction to PDEs, I highly recommend getting Farlow s text. It is easy to understand, and. bonus: It s really cheap on Amazon!! S.J. Farlow (1993). Partial Differential Equations for Scientists and Engineers. New York, NY: John Wiley & Sons (Did I mention cheap? $9.62 new on Amazon) Reading assignments: I will frequently assign readings from this list (especially the red ones) via Blackboard Learn!! Selected PDFs will be available for you to download on the class website. Additional All-purpose math textbooks: 1) G. Strang (1986). Introduction to Applied Mathematics. Wellesley, MA: Wellesley-Cambridge Press 2) K.F. Riley, M.P. Hobson, and S.J. Bence (2006). Mathematical Methods for Physics and Engineering: A Comprehensive Guide (3 rd ed.). Cambridge, UK: Cambridge University Press. Paperback 3) M. Boas (2006). Mathematical Methods in the Physical Sciences (3 rd ed.). Hoboken, NJ: John Wiley & Sons 4) E. Kreyszig (2011). Advanced Engineering Mathematics (10 th ed.). Hoboken, NJ: John Wiley & Sons PDE / ODE books: 1) S.J. Farlow (1993). Partial Differential Equations for Scientists and Engineers. New York, NY: John Wiley & Sons 2) S. Salsa (2008). Partial Differential Equations in Action: From Modelling to Theory. New York, NY: Springer-Verlag 3) J. D. Logan (2006): Applied Mathematics (3 rd ed.). Hoboken, NJ: John Wiley & Sons 4) S. I. Rubinow (1975). Introduction to Mathematical Biology. Hoboken, NJ: John Wiley & Sons 5) J.D. Murray (2003). Mathematical Biology II. Spatial Models and Biomedical Applications (3 rd ed.) New York, NY: Springer 6) L. Edelstein-Keshet (2005). Mathematical Models in Biology. Philadelphia, PA: SIAM 7) N.F. Britton (2003). Essential Mathematical Biology. London, UK: Springer Heat transfer / fluid mechanics 1) T.L. Bergman, A.S. Lavine, F.P. Incropera, and D.P.DeWitt (2011). Fundamentals of Heat and Mass Transfer. Hoboken, NJ: John Wiley & Sons 2) I.G. Currie (2012). Fundamentals Mechanics of Fluids (4 th ed). New York, NY: Marcel Dekker

5 Linear algebra (Sadun has a have very strong presentation of PDEs that connects back to linear algebra) 1) G. Strang (2009). Introduction to Linear Algebra (4 th ed.). Wellesley, MA: Wellesley-Cambridge Press. 2) G. Strang (2005). Linear algebra and its Applications (4 th ed.). Boston, MA: Cengage Learning 3) C.D. Meyer (2000). Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics (Siam) 4) S. Axler (1997). Linear Algebra Done Right (2 nd ed.). New York, NY: Springer-Verlag 5) L. Sadun (2008). Applied Linear Algebra: The Decoupling Principle (2 nd ed.). Providence, RI: American Mathematical Society Numerical methods (PDE-related): 1) L.N. Trefethen and D. Bau III (1997). Numerical Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics (Siam) 2) R. LeVeque (2007). Finite Difference methods for Ordinary and Partial Differential Equations: Steady-State and Timedependent Problems Philadelphia, PA: Society for Industrial and Applied Mathematics (Siam) 3) J.W. Thomas (1995). Numerical Partial Differential Equations: Finite Difference methods. New York, NY: Springer- Verlag 4) J.W. Thomas (1999). Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations. New York, NY: Springer-Verlag 5) Y. Saad (2003). Iterative Methods for Sparse Linear Systems (2 nd ed.). Philadelphia, PA: Society for Industrial and Applied Mathematics (Siam) 6) R. LeVeque (2005). Numerical Methods for Conservation Laws (2 nd ed.). Basel, Switzerland: Birkhauser 7) C. Johnson (2009). Numerical Solutions of Partial Differential Equations by the Finite Element Method. Mineola, NY: Dover